MG14 - Talk detail

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The Loop Quantum Gravity framework has been mostly studied in the case of a zero vanishing cosmological constant. A Loop Quantum Gravity model with a cosmological constant is not well understood even in the 3d toy model case. For the 3d case, several approaches to define a model of quantum gravity exist and in the case of a non-vanishing cosmological constant, the models such as the Turaev-Viro spin foam model or the Chern-Simons model are written in terms of a quantum group. To reconcile the Loop Quantum Gravity approach with these models, we deform the Loop Quantum Gravity framework using quantum groups or Poisson-Lie groups. Following this line, a topological model with a Hamiltonian constraint for 3d gravity with a cosmological constant has been defined. The Hamiltonian constraint can be solved and the solutions can be related to the Turaev-Viro spin foam amplitude. Moreover, geometric observables for quantum hyperbolic geometries have been defined. I am going to present an overview of this program consisting in introducing a cosmological constant in the 3d Loop Quantum gravity framework and will comment the 4d case.

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